3-2 Triangles - Grade 9 Math
Key Concepts
1. Definition of 3-2 Triangles
A 3-2 triangle is a special type of right triangle where the ratio of the lengths of the sides is 3:4:5. This means that if the shortest side (opposite the smallest angle) is 3 units, the next side (opposite the next largest angle) is 4 units, and the hypotenuse (opposite the right angle) is 5 units.
2. Properties of 3-2 Triangles
The properties of 3-2 triangles include:
- The sides are in the ratio 3:4:5.
- The angles are 36.87°, 53.13°, and 90°.
- It is a right triangle, meaning one of the angles is 90°.
3. Applications of 3-2 Triangles
3-2 triangles are commonly used in geometry and trigonometry. They are particularly useful in construction and navigation, where precise measurements are crucial.
Detailed Explanation
Example 1: Using the 3-2 Ratio
Problem: If the shortest side of a 3-2 triangle is 6 units, find the lengths of the other two sides.
Solution:
Given the ratio 3:4:5, and the shortest side is 6 units:
The next side is \(6 \times \frac{4}{3} = 8\) units.
The hypotenuse is \(6 \times \frac{5}{3} = 10\) units.
So, the sides are 6 units, 8 units, and 10 units.
Example 2: Finding Angles in a 3-2 Triangle
Problem: Find the angles in a 3-2 triangle.
Solution:
Using trigonometric ratios:
The smallest angle (\(\theta\)) can be found using \(\sin(\theta) = \frac{3}{5}\).
\(\theta = \sin^{-1}(\frac{3}{5}) \approx 36.87°\)
The next angle is \(90° - 36.87° = 53.13°\)
So, the angles are 36.87°, 53.13°, and 90°.
Analogies for Clarity
Building a Frame
Think of a 3-2 triangle as a frame for a picture. The shortest side is the base, the next side is the height, and the hypotenuse is the diagonal support. This frame is stable because the sides are in the 3:4:5 ratio, ensuring the angles are precise.
Navigating by Compass
In navigation, a 3-2 triangle can be used to determine distances and directions. For example, if you know the shortest distance (3 units) and need to find the longest distance (5 units), you can use the 3:4:5 ratio to calculate the intermediate distance (4 units) and ensure accurate navigation.
Conclusion
3-2 triangles are a fundamental concept in geometry and trigonometry, providing a simple yet powerful tool for solving problems involving right triangles. By understanding the ratio and properties of 3-2 triangles, you can apply this knowledge to various real-world scenarios, from construction to navigation.